On Unsheared Tetrads (original) (raw)

Generalization of the tetrad representation theorem

1993

The tetrad representation theorem, due to Spirtes, Glymour, and Scheines (1993), gives a graphical condition necessary and su cient for the vanishing of tetrad di erences in a linear correlation structure. This note simpli es their proof and generalizes the ...

On quasigroups rich in associative triples

Discrete Mathematics, 1983

Let G be a group and G(*) a quasigroup on the ~.~ame underlying set. Let dist(G, G(*)) denote the number of pairs (x,y)eG 2 such that xy~x*y. For a finite quasig4"oup Q, n = card(Q), let t = dist(Q) = min dist(G, Q), where G ruu-~ through all groups with the same underlying set, and s = s(Q) the number of non-associative triples. Then 4tn-2t 2 -24t ~ s ~< 4m. If 1~,.~<3n2/32, then 3m<s holds as well. Let n~>168 be an even integer and let tr=mins(Q), where Q runs through all non-associative quasigroups of order n. Then tr= 16n -64.

On radicals and products

Pacific Journal of Mathematics, 1985

An Abelian group G is called cotorsion-free if 0 is the only pure-injective subgroup contained in G. If G is a cotorsion-free Abelian group, we construct a slender, fr^-free Abelian group A such that Hom(Λ, G) = 0. This will be used to answer some questions about radicals and torsion theories of Abelian groups. 0. Introduction. In this paper we will consider torsion free abelian groups from I. Kaplansky's point of view: "In this strange part of the subject anything that can conceivably happen actually does happen", cf. [K, p. 81]. This statement which is supported by classical results holds in an even more spectacular sense which was not expected at this time. There are many results on torsion free abelian groups which are undecidable in ZFC, the axioms of Zermelo-Frankel set theory including the axiom of choice. The first suφrising result of this kind after years of stagnation was Shelah's solution of the famous Whitehead problem [SI]. In this paper Shelah also constructed for the first time arbitrarily large indecomposable abelian groups, thus improving classical results of S. Pontrjagin, R. Baer, I. Kaplansky, L. Fuchs, A. L. S. Corner and others, compare [Fu2, Vol. II] and [K]. Indecomposable abelian groups are necessarily cotorsion-free with only a few exceptions. These are the cyclic groups of prime power Z p «, the Prϋfer groups Z(/?°°), the group of rational numbers Q and the additive group J p of /?-adic integers. A group is called cotorsion-free if and only if it contains only the trivial cotorsion subgroup 0, cf. [GW1]. Remember that C is cotorsion (in the sense of K. H. Harrison) if Ext z (Q, C) = 0. From simple properties of cotorsion groups we conclude that a group G is cotorsion-free if and only if G is torsion-free (Z p £G),reduced(Q£G)andJp£ G), reduced (Q £ G) and J p £G),reduced(Q£G)andJpt G for all primes p, cf. [GW1]. For countable groups cotorsion-free is the same as reduced and torsion-free. A. L. S. Corner's celebrated theorem indicates then that each ring with a countable and cotorsion-free additive structure is the endomorphism ring of some (cotorsion-free) abelian group, cf. [Ful, Vol. II]. This result was extended by the authors [DG2] to arbitrary rings with cotorsion-free additive groups which are then realized on arbitrarily large cotorsion-free abelian groups. Using rings without non-trivial idempotents, indecomposable groups of 79 80 MANFRED DUGAS AND RUDIGER GOBEL any size can be obtained and the aforementioned result becomes a trivial consequence of [DG2]. However, using other elementary ring constructions this result supports Kaplansky's point of view in many aspects, e.g. there are many new different counter examples for I. Kaplansky's test problems. Similar results which are in many cases even undecidable im ZFC have been derived in [DG1], [EM], [Me], [DH1] and others. One of the questions "close" to results undecidable in ZFC is related with "rigid systems". A class {A^ i e /} of abelian groups is semi-rigid if Hom(v4 /? Aj) Φ 0 Φ Hom(Aj, A t) implies i = j for any /, j e /. This class is rigid if already Hom(yl 2 , Aj) Φ 0 implies i = j. The class is proper if / is not a set. M. Dugas and S. Herden [DH1] constructed proper rigid classes of (indecomposable) abelian groups using GδdePs axiom of constructibility V = L. Such a result cannot be expected in ZFC alone as follows from the Vopenka principle. However, at least semi-rigid proper classes exist in ZFC as recently shown by R. Gόbel and S. Shelah [GS]. This result is based on a construction of arbitrarily large cotorsion-free abelian groups A with the property that U = A for any subgroup U c A with \U\ = \A\ and A/U cotorsion free. All these constructions are highly sophisticated using transfinite induction on generating elements. The very heart of this paper is a similar kind of result based on a much simpler construction. Due to the elementary construction of the groups (4.2) we are able to pose stronger conditions on their structure, which allow us to answer some open problems and give new solutions to some already settled problems. These extra conditions are the properties N 1-free and slender. A group is called S Γ free if all its countable subgroups are free. The most popular non-free S 1-free groups are products Z* of the integers, in particular the Baer-Specker group Z s°. The proof that Z*° is fc^-free and not free is due to R. Baer and E. Specker, cf. [Ful, Vol. I]. We will use R. J. Nunke's well-known characterization of slender groups as a definition. Hence a group is slender if and only if it is cotorsion free and if it does not contain a copy of the Baer-Specker group. Then we have the following quite powerful

On Non-Cayley Tetravalent Metacirculant Graphs

Graphs and Combinatorics, 2002

In connection with the classification problem for non-Cayley tetravalent metacirculant graphs, three families of special tetravalent metacirculant graphs, denoted by U 1 , U 2 and U 3 , have been defined . It has also been shown [11] that any non-Cayley tetravalent metacirculant graph is isomorphic to a union of disjoint copies of a non-Cayley graph in one of the families U 1 , U 2 or U 3 . A natural question raised from the result is whether all graphs in these families are non-Cayley. We have proved recently in [12] that every graph in U 2 is non-Cayley. In this paper, we show that every graph in U 1 is also a connected non-Cayley graph and find an infinite class of connected non-Cayley graphs in the family U 3 .

On torsion torsionfree triples

2007

A mi familia y amigos. A Patri. Contents Introducción (Spanish) i 65 4.1.1. Motivation 65 4.1.2. Outline of the chapter 65 i ii CONTENTS INTRODUCCIÓN (SPANISH) iii 3) Existen isomorfismos de anillos (DA)(P, P ) ∼ = C y (DA)(Q, Q) ∼ = B. 4) (DA)(P [n], Q) = 0 para todo n ∈ Z. 5) Si M es un complejo de D − A tal que (DA)(P [n], M ) = 0 y (DA)(Q[n], M ) = 0 para todo n ∈ Z, entonces M = 0.

Tetrads of lines spanning PG(7,2)

Bulletin of the Belgian Mathematical Society Simon Stevin, 2012

Our starting point is a very simple one, namely that of a set L_4 of four mutually skew lines in PG(7,2): Under the natural action of the stabilizer group G(L_4) < GL(8,2) the 255 points of PG(7,2) fall into four orbits omega_1, omega_2, omega_3 omega_4; of respective lengths 12, 54, 108, 81: We show that the 135 points in omega_2 \cup omega_4 are the internal points of a hyperbolic quadric H_7 determined by L_4; and that the 81-set omega_4 (which is shown to have a sextic equation) is an orbit of a normal subgroup G_81 isomorphic to (Z_3)^4 of G(L_4): There are 40 subgroups (isomorphic to (Z_3)^3) of G_81; and each such subgroup H < G_81 gives rise to a decomposition of omega_4 into a triplet of 27-sets. We show in particular that the constituents of precisely 8 of these 40 triplets are Segre varieties S_3(2) in PG(7,2): This ties in with the recent finding that each Segre S = S_3(2) in PG(7,2) determines a distinguished Z_3 subgroup of GL(8,2) which generates two sibling copies S'; S" of S.

A Family of Tetravalent Half-transitive Graphs

2020

In this paper, we introduce a new family of graphs, Gamma(n,a)\Gamma(n,a)Gamma(n,a). We show that it is an infinite family of tetravalent half-transitive Cayley graphs. Apart from that, we determine some structural properties of Gamma(n,a)\Gamma(n,a)Gamma(n,a).

Discrete Mathematics 35 (198 1) 25-38 North-Holland Publishing Company

An explicit construction is given Which produces all the proper fiats and the'Tutte ~lynomial of a geometric lattice (or, more generally, a matroid) when only the hypeq&nes are know. A further construction explicitly ccrlculatzs the polychromate (a genera&&on of the Tutte polynomial) for a graph from its vertex-deleted subgrqhs.

Triply Existentially Complete Triangle-Free Graphs

Journal of Graph Theory, 2014

A triangle-free graph G is called k-existentially complete if for every induced k-vertex subgraph H of G, every extension of H to a (k + 1)-vertex triangle-free graph can be realized by adding another vertex of G to H. Cherlin [11, 12] asked whether k-existentially complete triangle-free graphs exist for every k. Here, we present known and new constructions of 3existentially complete triangle-free graphs.

Arc-transitive cycle decompositions of tetravalent graphs

Journal of Combinatorial Theory, Series B, 2008

A cycle decomposition of a graph Γ is a set C of cycles of Γ such that every edge of Γ belongs to exactly one cycle in C. Such a decomposition is called arc-transitive if the group of automorphisms of Γ that preserve C setwise acts transitively on the arcs of Γ. In this paper, we study arc-transitive cycle decompositions of tetravalent graphs. In particular, we are interested in determining and enumerating arc-transitive cycle decompositions admitted by a given arc-transitive tetravalent graph. Among other results we show that a connected tetravalent arc-transitive graph is either 2-arc-transitive, or is isomorphic to the medial graph of a reflexible map, or admits exactly one cycle structure.

CSP dichotomy for special triads

Proceedings of the American Mathematical Society, 2009

For a fixed digraph G, the Constraint Satisfaction Problem with the template G, or CSP(G) for short, is the problem of deciding whether a given input digraph H admits a homomorphism to G. The dichotomy conjecture of Feder and Vardi states that CSP(G), for any choice of G, is solvable in polynomial time or NP-complete. This paper confirms the conjecture for a class of oriented trees called special triads. As a corollary we get the smallest known example of an oriented tree (with 33 vertices) defining an NP-complete CSP(G).

Tetravalent -transitive graphs of order

Discrete Mathematics, 2009

Let s be a positive integer. A graph is s-transitive if its automorphism group is transitive on s-arcs but not on (s+1)-arcs, and 1 2-arc-transitive if its automorphism group is transitive on vertices, edges but not on arcs. Let p be a prime. Feng et al. [Y.-Q. Feng, K.S. Wang, C.X. Zhou, Tetravalent half-trasnitive graphs of order 4p, European J. Combin. 28 (2007) 726-733] classified tetravalent 1 2-arc-transitive graphs of order 4p. In this article a complete classification of tetravalent s-transitive graphs of order 4p is given. It follows from this classification that with the exception of two graphs of orders 8 or 28, all such graphs are 1-transitive. As a result, all tetravalent vertex-and edge-transitive graphs of order 4p are known.

Graph theory discussion and some new results

2014

A(Tn) Alternate triangular snake A(QSn) Alternate quadrilateral snake Bn Book graph Bn,n Bistar graph |B| Cardinality of set B Cn Cycle with n vertices CHn Closed Helm D2(G) Shadow graph of G D fn Double fan graph DTn Double triangular snake DA(Tn) Double alternate triangular snake DA(QSn) Double alternate quadrilateral snake d(v) or dG(v) Degree of a vertex v of graph G E(G) Edge set of graph G fn Fan graph Fn Friendship graph Fln Flower graph Hn Helm graph G Graph G×H Cartesian product of graphs G and H G(Tp)H Tensor product of graphs G and H Gn Gear graph G2 Square of graph G G3 Cube of graph G

A note on some chain conditions

Archiv der Mathematik, 2008

We answer a question raised by Othman Echi: Is an E1 (resp., a C1) ring an E (resp., a C) ring? We construct a C1 (thus E1) ring which is not an E2 (thus not a C2) ring.

On universally catenarian pairs

Journal of Pure and Applied Algebra, 2008

If 1 ≤ n < ∞ and R ⊆ S are integral domains, then (R, S) is called an n-catenarian pair if for each intermediate ring T (that is each ring T such that R ⊆ T ⊆ S) the polynomial ring in n indeterminates, T[n] is catenarian. This implies that (R, S) is m-catenarian for all m < n. The main purpose of this paper is to prove that 1-catenarian and universally catenarian pairs are equivalent in several cases. An example of a 1-catenarian pair which is not 2-catenarian is given.

On the CO-\u3cem\u3eP\u3csub\u3e3\u3c/sub\u3e\u3c/em\u3e-Structure of Perfect Graphs

2005

Let F be a family of graphs. Two graphs G 1 = (V 1 , E 1), G 2 = (V 2 , E 2) are said to have the same F-structure if there is a bijection f : V 1 → V 2 such that a subset S induces a graph belonging to F in G 1 if and only if its image f (S) induces a graph belonging to F in G 2. We characterize those graphs which have the same {P 3 , P 3 }-structure, or the same {K 3 , K 3 }-structure. This characterization shows that graph H is perfect if and only if it has the {P 3 , P 3 }-structure of some perfect graph G. In proving the main result, we need and prove the following result, which is of independent interest: If a graph J is claw-free and co-claw-free, then either (i) J has at most nine vertices, or (ii) every component of J is a path or a hole, or (iii) every component of J is a path or a hole.

On quasigroups with few associative triples

Designs, Codes and Cryptography, 2012

There are numerous application of quasigroups in cryptology. It turns out that quasigroups with the relatively small number of associative triples can be utilized in designs of hash functions. In this paper we provide both a new lower bound and a new upper bound on the minimum number of associative triples over quasigroups of a given order.

On Walkup’s class and a minimal triangulation of

Discrete Mathematics, 2011

For d ≥ 2, Walkup's class K(d) consists of the d-dimensional simplicial complexes all whose vertex-links are stacked (d − 1)-spheres. Kalai showed that for d ≥ 4, all connected members of K(d) are obtained from stacked d-spheres by finitely many elementary handle additions. According to a result of Walkup, the face vector of any triangulated 4-manifold X with Euler characteristic χ satisfies f 1 ≥ 5f 0 − 15 2 χ, with equality only for X ∈ K(4). Kühnel observed that this implies f 0 (f 0 − 11) ≥ −15χ, with equality only for 2-neighborly members of K(4). Kühnel also asked if there is a triangulated 4-manifold with f 0 = 15, χ = −4 (attaining equality in his lower bound). In this paper, guided by Kalai's theorem, we show that indeed there is such a triangulation. It triangulates the connected sum of three copies of the twisted sphere product S 3 S 1. Because of Kühnel's inequality, the given triangulation of this manifold is a vertex-minimal triangulation. By a recent result of Effenberger, the triangulation constructed here is tight. Apart from the neighborly 2manifolds and the infinite family of (2d + 3)-vertex sphere products S d−1 × S 1 (twisted for d odd), only fourteen tight triangulated manifolds were known so far. The present construction yields a new member of this sporadic family. We also present a self-contained proof of Kalai's result.